This happened a few times that I can think of - I’m sure it probably happened even more often, but here are some examples from my areas of expertise:

First of all this was exactly the status of the King of England himself after 1066 - the kings were the Dukes of Normandy, which was pretty much de facto independent but was technically a vassal of the King of France. So William I and subsequent kings ruled England as an independent kingdom, but they still owed homage/service/taxes as the Duke of Normandy to his liege lord, the king of France. The same was true when the County of Anjou and the Duchy of Aquitaine came under the control of the kings of England. If everyone strictly followed the customary rules, the king of England owed money and military service to the king of France. Of course it doesn’t really work out that way…like, what if the king of France demands the count of Anjou provide military service somewhere in France, and since the count of Anjou is also the king of England, he just says no? Especially if the military expedition is against some other land that falls within the king of England’s interest?

Another example is Peter of Dreux, who was just a minor noble with some territory in the County of Dreux - he wasn’t even the count of Dreux himself, the count was his brother Robert. So in the chain of vassals, he owed homage to his own brother the count of Dreux, and the count of Dreux owed homage to the Count of Champagne and the count of Champagne was a vassal of the king of France. But then Peter married the Duchess of Brittany, who was well out of his league, since Brittany (unlike Champagne or Normandy) was never actually a part of France, it was independent both legally and in practice. When Peter’s wife died, he ruled Brittany on his own (although technically he was only regent for his son). Brittany didn’t owe anything to the king of France, service, taxes, or otherwise, but it was always dominated either by France or England, and in Peter’s time France was the dominant force there. Peter could fight alongside the king if he wanted to, and sometimes he died; if not, the king of France could invade and fight against him, which also happened sometimes. In this case, when Peter’s son John was old enough to rule on his own, Peter was no longer lord of anything except his lands back in the county of Dreux.

Another example is Thibaut IV, Count of Champagne (and one of Peter of Brittany’s lords!), who also inherited the Kingdom of Navarre through his mother. Champagne, like Normandy, was theoretically a vassal of France, but it was definitely de facto independent. King Louis VIII once asked for Thibaut’s military service during the Albigensian Crusade, and Thibaut spent exactly 40 days on campaign, as required, no more, no less - and there wasn’t much the king could do about it. So here there was the weird situation where Thibaut is an independent king, but also a vassal of the king of France, and Peter was an independent Duke, as well as vassal of France and more directly a vassal of Champagne.

Thibaut and Peter and King Louis IX of France all went on crusade together as well - so who was lord of whom? Or were they all on crusade as independent rulers?

Also, what happened if the two territories were too far away to rule at the sometime? Champagne and Navarre were pretty far away but not too far. Any further apart and things might get difficult. Baldwin IX of Flanders took part in the Fourth Crusade and ended up becoming the Latin Emperor of Constantinople - and he was rather ineffectual at ruling both Constantinople and Flanders at the same time. When he was killed in battle in 1205 he had different successors in both places, so that solved that problem.

Another way to prevent problems was to simply resign the lesser position. For example, Fulk V of Anjou became King of Jerusalem in 1131, but he gave up his title in Anjou to his son Geoffrey (which, incidentally, is how Anjou ended up ruled by the kings of England - Geoffrey’s son became King Henry II of England).

One last example is Richard, Earl of Cornwall, who was also elected King of Germany in 1257 - but he hardly ever visited Germany so there wasn’t really a conflict of interests there.

So in short, there was no single rule, and whenever this happened they tended make things up as they went along. But you could certainly rule both places at the same time, owing different services taxes on the land where you’re the vassal of a different king, while also ruling as a monarch on your own territory.

Here are a few sources if you'd like to read more about these possibilities:

Theodore Evergates, The Aristocracy in the County of Champagne, 1100-1300 (University of Pennsylvania Press, 2007)

Sidney Painter, Scourge of the Clergy: Peter of Dreux, Duke of Brittany (Johns Hopkins University Press, 1937)

Elizabeth M. Hallan, Capetian France, 987-1328 (Longman, 1980, repr. 1990)